Does the data fit the claimed distribution?
The goodness-of-fit test compares counts you observed against counts a hypothesis predicts: a fair die (equal sixths), last year's sales mix, Mendel's 9:3:3:1. Each category contributes (O − E)²/E to the statistic: big misses on small expected counts hurt most, because relative to what was promised, they are the loudest disagreements.
Sixty rolls of a die: 20, 8, 10, 6, 9, 7 against 10 expected per face. The 20 alone contributes (20−10)²/10 = 10 of the total χ² = 13.0 (df 5, p = 0.0234). At 0.05 the verdict is: this die's rolls do not fit "fair". Read the contributions table before the p-value. One swollen category is a different story from a general misfit, and the statistic keeps them separate for you.
The expected-count rule
The chi-square approximation needs expected counts of roughly 5 or more per category. Starved categories should be merged with neighbours (fewer, fatter categories), or, for very small samples, an exact multinomial calculation used instead.
Feeding percentages or proportions into a chi-square test. The statistic is built from counts; 60 rolls and 600 rolls with identical proportions are very different evidence, and only the counts know it.
Frequently asked questions
Why do small expected counts break the test?
The chi-square curve approximates the statistic's distribution, and the approximation is poor when a category's expected count is tiny: a single observation swings its contribution wildly.
Can I test against last year's proportions?
Yes. Choose the proportions mode and enter last year's shares. The calculator converts them to expected counts from your total.